The publisher asks me if I want a CC-BY, CC-BY-NC, CC-BY-ND of CC-BY-ND-NC. What does that mean exactly?

These are various Creative Commons licences. 3 common variants are briefly described below. Visit the Creative Commons website or the Terms and conditions of use page for more information.

CC-BY
CC-BY is the Creative Commons Attribution licence. You thereby permit everyone to distribute, copy and modify your work, provided that they credit the author. Modification may refer to a translation, an abridged version, a summary or other type of conversion into a different form.

CC-BY-ND
The addition ND means 'no derivatives', so no derivative works are permitted. A CC-BY-ND licence permits people to distribute and copy your work, provided the work remains in its original and provided that they credit the author. However, it doesn't permit them to modify your work. Your work may therefore not be translated, modified or adapted into any other form without your permission. The advantage of this is that no one can 'ruin' your work. A disadvantage is that if someone wants to translate, shorten or save your work in another format, this person must first request your (and the co-authors') permission (for as long as your article is subject to copyright).

CC-BY-NC
The addition NC means 'non-commercial use', so no commercial use is permitted. A CC-BY-NC licence permits people to distribute, copy and modify your work, provided that they credit the author. However, it doesn't permit them to use your work for commercial purposes. Your work may therefore not be used in a commercial sense without your (and your co-authors') permission.

In some cases, the commercial use is clear: a company 'sells' your work (while it is available elsewhere for free). However, the problem is that commercial use also has a grey area. There is no clear definition of commercial use. Is indexing something and making it available via Google commercial use? Is using your work in a private collaboration commercial use? Is reproducing it at printing cost? Consequently, we advise against using NC.

A special note of caution on the use of NC and/or ND licenses by publishers
When selecting a CC BY-NC or ND license for an article and signing the so-called “License to Publish” agreement prior to publication, authors commonly (but unknowingly) transfer the exclusive right to manage the license to their publisher. In this way, the right to commercially reuse or make adaptations is no longer in the hands of the author. Not all publishers communicate this clearly to authors. Read more about this issue in the 'Guide to Creative Commons for Scholarly Publications and Educational Resources', p. 10.